Recreate the diagram I drew in lectures. In this diagram, the Earth is expressed as a "big rectangle", or 2-D map. The lines at +/- 23.5° (i.e., 23.5°N and 23.5°S) latitude (the tropics) are marked. The declination of the Sun is drawn in the middle of the map, so that it moves from +/-23.5° over the course of a year and crosses 0° on the Equinoxes. Remember, 90° from your position on Earth in this diagram is your horizon and 0° from your position is zenith (directly over your head).
Complex Worked Example
You live at 83.5°N. How far does the Earth get above or below the horizon at local noon and local midnight on (a) December 21 (b) March 21 (c) June 21?
(a) December 21, step-by step
(i) What is the declination of the Sun on December 21? 23.5°S.
(ii) This means that the Sun is above a latitude of 23.5°S on the Earth's surface.
(iii) During the day, how far is the Sun at 23.5°S from 83.5°N? 83.5+23.5 = 107°
(iv) The zenith is 90° from the horizon. Therefore 107° from zenith is 17° below the horizon.
(v) During the night, how far is the Sun at 23.5°S from 83.5°N (it's night so we go up to the pole and down the other side of the Earth). 90-83.5 = 6.5. So it's 6.5° to the pole, 6.5° back down to 83.5°N on the daytime side the Earth, then the previous distance from the daytime calculation down to the Sun. 6.5 + 6.5 + (83.5+23.5) = 6.5 + 6.5 + 107 = 120°
(vi) The zenith is 90° from the horizon. Therefore 120° from zenith is 30° below the horizon.
(vii) If you live at 83.5°N the Sun is 17° below the horizon at local noon. It is 30° below the horizon at local midnight. Therefore, the Sun never rises at any time of the day on December 21 if you live at 83.5°N.
(b) March 21, step-by step
(i) What is the declination of the Sun on March 21? 0°N
(ii) This means that the Sun is above a latitude of 0°N on the Earth's surface.
(iii) During the day, how far is the Sun at 0°N from 83.5°N? 83.5+0 = 83.5°
(iv) The zenith is 90° from the horizon. Therefore 83.5° from zenith is 6.5° above the horizon.
(v) During the night, how far is the Sun at 0°N from 83.5°N (it's night so we go up to the pole and down the other side of the Earth). 90-83.5 = 6.5. So it's 6.5° to the pole, 6.5° back down to 83.5°N on the daytime side the Earth, then the previous distance from the daytime calculation down to the Sun. 6.5 + 6.5 + (83.5+0) = 6.5 + 6.5 + 83.5 = 96.5°
(vi) The zenith is 90° from the horizon. Therefore 96.5° from zenith is 6.5° below the horizon.
(vii) If you live at 83.5°N the Sun is 6.5° above the horizon at local noon. It is 6.5° below the horizon at local midnight. Therefore, the Sun is up during the day but down at night on March 21 (or in fact, either Equinox) if you live at 83.5°N. In fact, because it's the Equinox, it's up for 12 hours during the day and 12 hours at night.
(c) June 21, step-by step
(i) What is the declination of the Sun on June 21? 23.5°N
(ii) This means that the Sun is above a latitude of 23.5°N on the Earth's surface.
(iii) During the day, how far is the Sun at 23.5°N from 83.5°N? 83.5-23.5 = 60°
(iv) The zenith is 90° from the horizon. Therefore 60° from zenith is 30° above the horizon.
(v) During the night, how far is the Sun at 23.5°N from 83.5°N (it's night so we go up to the pole and down the other side of the Earth). 90-83.5 = 6.5. So it's 6.5° to the pole, 6.5° back down to 83.5°N on the daytime side the Earth, then the previous distance from the daytime calculation down to the Sun. 6.5 + 6.5 + (83.5-23.5) = 6.5 + 6.5 + 60 = 73°
(vi) The zenith is 90° from the horizon. Therefore 73° from zenith is 17° above the horizon.
(vii) If you live at 83.5°N the Sun is 30° above the horizon at local noon. It is 17° above the horizon at local midnight. Therefore, the Sun is up during both the day and the night on June 21 if you live at 83.5°N. The Sun never sets.
Questions
1. You live at 40°N. How far does the Sun get above the horizon at local noon on December 21?
2. You live at 40°N. How far does the Sun get above the horizon at local midnight on June 21?
3. You live at 40°N. How far does the Sun get above the horizon at local noon on March 21?
4. You live at 40°N. How far does the Sun get above the horizon at local midnight on September 21?
5. You live at 66.5°N. How far does the Sun get above the horizon at local midnight on June 21?
6. You live at 66.5°N. How far does the Sun get above the horizon at local noon on June 21?
7. You live at 0°N. How far does the Sun get above the horizon at local noon on June 21?
8. You live at 0°N. How far does the Sun get above the horizon at local midnight on March 21?
9. You live at 23.5°N. How far does the Sun get above the horizon at local noon on the September 21?
10. You live at 23.5°S. How far does the Sun get above the horizon at local noon on June 21?
Answers
Scroll a long way down the page for each answer
1. 26.5° above the horizon
2. 26.5° below the horizon
3. 50° above the horizon
4. 50° below the horizon
5. 0° on the horizon (just up at night)
6. 47° above the horizon
7. 66.5° above the horizon (close to zenith...you live in the tropics)
8. 90° below the horizon (and it would be 90° above the horizon, or right at zenith during the day)
